The energy distribution of conduction band states due to photoionisation correspond to near delta functions at the energy corresponding to the band locations at the momenta at which transition occurred. Many scattering events then take place which change the carrier distribution towards and away from equilibrium. Elastic scattering leads to momentum relaxation and will distribute the carriers more evenly across the Brillouin zone and inelastic scattering leads to energy relaxation and will ultimately bring the carriers back towards equilibrium with the lattice [1]}. Given no driving field, collisional energy losses average out such that the distribution thermalises through screened Coulomb interaction carrier-carrier collisions to an effective temperature much hotter than the lattice. Electron-phonon coupling causes the phonon subsystem to react to the non-equilibrium distribution of electrons generating longitudinal optical and acoustic phonons which are emitted from carriers transferring energy to the lattice [2]}. The carrier-phonon coupling strength in the conduction band is much stronger than in the valence band, therefore the relaxation of holes takes place over a longer timescale [3]}. These scattering events lead to cooling of the carrier density to the Fermi-Dirac distribution in equilibrium with the lattice. Under a driving electric field, carriers can absorb photons if their oscillation is knocked out of phase with the incoming field [4]}. This is known as reverse Bremsstrahlung or carrier-phonon-photon scattering which leads to an efficient energy gain of carriers, skewing the distribution away from equilibrium. The photoionisation and reverse Bremsstrahlung processes compete with the phonon and carrier scattering leading to a skewed thermal distribution.
We approximate the rates of each of these terms as a function of space, time and energy by using a mixture of collision time approximations, Drude theory and energy and momentum conservation [5]} [6]}. The average carrier-carrier collision time is calculated using the change to the carrier mobility as a function of carrier density under the assumption that each scattering event randomises the motion [7]} given by the equation.
\( \begin{split}\tau _{cc}(r,z,t)=\frac{m^{\prime } a T(r,z,t)^\frac{2}{3}}{e N_{carr}(r,z,t)}\\\times \frac{1}{\ln (1 + b T(r,z,t)^2 (1\times 10^{-4} \cdot N_{carr}(r,z,t))^{-\frac{2}{3}})}\end{split}\qquad \mathrm {(S4)}\) 
